Calc. Var. (2013) 46:325–351 DOI 10.1007/s00526-011-0484-x Calculus of Variations Weakly coupled nonlinear Schrödinger systems: the saturation effect Liliane de Almeida Maia · Eugenio Montefusco · Benedetta Pellacci Received: 21 April 2011 / Accepted: 10 November 2011 / Published online: 26 January 2012 © Springer-Verlag 2012 Abstract We study the existence of solutions for a class of saturable weakly coupled Schrödinger systems. In most of the cases we show that least energy solutions have neces- sarily one trivial component. In addition sufficient conditions for the existence of a solution with both positive components are found. Mathematics Subject Classification (2010) 35J20 · 35Q55 · 35Q60 1 Introduction Nonlinear optic is a matter of great interest in technologies research, since it provides good knowledge of the transmission of light at high velocity. In this context, spatial optical solitons have recently attracted much attention due to their good properties: they are solitary waves, that maintain their shape during propagation (the reader can see the book [1] for a review on optical solitons). The passage of a ray along different materials induces several nonlinear effects. The single Schrödinger equation well represents the auto-interaction of the beam, but it does not take into account possible interactions with the material. For instance, the passage of a light beam with two orthogonally polarized components along an optical fiber produces the birefrin- gence effect: the decomposition of the ray into two. Then the model of a single Schrödinger Communicated by A. Malchiodi. L. de Almeida Maia Departmento de Matemática, Universidade de Brasília, 70910-900 Brasilia, Brazil e-mail: lilimaia@unb.br E. Montefusco (B ) Dipartimento di Matematica, Sapienza Università di Roma, Piazzale Aldo Moro 5, 00185 Roma, Italy e-mail: montefusco@mat.uniroma1.it B. Pellacci Dipartimento di Scienze Applicate, Università degli Studi di Napoli Parthenope, Centro Direzionale Isola C4, 80143 Napoli, Italy e-mail: pellacci@uniparthenope.it 123